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A horizontal line intersects $AB$ at $T$ and $AC$ at $U$, forming $\triangle ATU$. If the area of $\triangle ATU$ is $13.5$, determine the equation of the horizontal line.

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project. Solutions are the publisher's, linked not copied.